Determinant

Imagine the area of parallelogram created by the basis of a standard vector space, like ${\mathbb{R}}^2$. Now apply a linear transformation $A$ to that vector space. The new area of the new parallelogram has been scaled by a factor of the determinant. $S$ is the parallelopiped.

$$area(T(S))=|det(A)|\cdot area(S)$$

You can also just think of it as the area of the parallelogram spanned by the columns of a matrix R^3 and beyond → parallelopiped and volume (assume n by n matrix because we only know how to find determinants for square matrices)

You can also get the area of S by using the determinant of the matrix created by the vectors that span S, i.e. $|\vec{a}\times \vec{b}|=area(S)⟹|det([\vec{a}\vec{b}])|=area(S)$ because you are shifting the standard basis vectors into the vector space dictated by S

Determinant Laws

• det(A) = 0 $⟺$ A is singular
  • det(A) $\ne$ 0 $⟺$ A is invertible
• det(Triangular) = product of diagonals
• det A = det $A^T$
• det(AB) = det A · det B
• $det(A^{-1})=\frac{1}{det(A)}$
• $det(kA)=k^{n}det(A)$

Determinant Post Row Operations

if A square:

• if adding rows to rows on A to get B then $detA=detB$
• if swapping rows in A to get B then $-detA=detB$
• if scaling one row of A by k, then $k\cdot det(A)$ = $det(B)$ Exactly the same for columns